L(s) = 1 | + (0.541 − 1.30i)2-s + (−1.64 − 0.541i)3-s + (−1.41 − 1.41i)4-s + (−1.59 + 1.85i)6-s + 3.29i·7-s + (−2.61 + 1.08i)8-s + (2.41 + 1.78i)9-s − 2.51i·11-s + (1.56 + 3.09i)12-s + 4.65i·13-s + (4.29 + 1.78i)14-s + 4i·16-s + 3.69i·17-s + (3.63 − 2.19i)18-s − 0.828·19-s + ⋯ |
L(s) = 1 | + (0.382 − 0.923i)2-s + (−0.949 − 0.312i)3-s + (−0.707 − 0.707i)4-s + (−0.652 + 0.758i)6-s + 1.24i·7-s + (−0.923 + 0.382i)8-s + (0.804 + 0.593i)9-s − 0.759i·11-s + (0.450 + 0.892i)12-s + 1.29i·13-s + (1.14 + 0.475i)14-s + i·16-s + 0.896i·17-s + (0.856 − 0.516i)18-s − 0.190·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.968065 + 0.0362796i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.968065 + 0.0362796i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.541 + 1.30i)T \) |
| 3 | \( 1 + (1.64 + 0.541i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.29iT - 7T^{2} \) |
| 11 | \( 1 + 2.51iT - 11T^{2} \) |
| 13 | \( 1 - 4.65iT - 13T^{2} \) |
| 17 | \( 1 - 3.69iT - 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 - 6.08T + 29T^{2} \) |
| 31 | \( 1 - 1.17iT - 31T^{2} \) |
| 37 | \( 1 + 1.92iT - 37T^{2} \) |
| 41 | \( 1 - 8.59iT - 41T^{2} \) |
| 43 | \( 1 + 6.01T + 43T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 - 4.59T + 53T^{2} \) |
| 59 | \( 1 + 2.51iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 3.29T + 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 - 16.4iT - 79T^{2} \) |
| 83 | \( 1 - 9.37iT - 83T^{2} \) |
| 89 | \( 1 - 5.03iT - 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.94408517709371240060889549605, −10.06904452940281698676490999828, −9.040608258796369860601602103811, −8.342432506027592002313514263626, −6.65515849959084352960273761160, −5.97565706210678355838234823662, −5.12777589453713050576278472753, −4.12802205199083457142145209520, −2.64264435077479216324767499882, −1.43819612774509355104409979434,
0.58758149524317795814131619814, 3.31239687485636005858848996874, 4.44720818619131293467525561805, 5.05692829818116707895014084135, 6.11061344942395000692540816941, 7.11839630905412990022545586646, 7.51873857225906112111321775289, 8.824406939683461585451889318177, 10.01859040074109062539602670670, 10.42690783468509596069093478517